Method and device of fault location

ABSTRACT

The present invention relates to a method for location of a fault utilizing unsynchronized measurements of three phase voltages and currents acquired at the line terminals without synchronization. Phasors for symmetrical components of the measured quantities are determined and used in the fault location algorithm. According to one embodiment, positive sequence phasors of post-fault quantities are used for estimation of the distance to fault and it is distinctive that such an estimation of a distance to fault is performed without essentially involving iterative techniques. In this embodiment, the fault location algorithm step is performed regardless of the fault type. In later steps, the type of fault may be obtained. According to another embodiment of the invention, at the occurrence of a fault, the type of fault is determined. If it is determined that it is not a three-phase balanced fault, negative sequence phasors are used for the estimation of the distance to fault. On the other hand, if it is a three-phase balanced fault, the incremental positive sequence phasors are used.

TECHNICAL AREA

[0001] The present method of fault location, which is presented here, isbased on utilizing unsynchronized measurements of three phase voltagesand currents acquired at the line terminals without synchronization.Phasors for symmetrical components of the measured quantities aredetermined and used in the fault location algorithm. Basically, positivesequence phasors of post-fault quantities are used for estimation of thedistance to fault and it is distinctive that such an estimation of adistance to fault is performed without involving iterative techniques.

BACKGROUND OF THE INVENTION

[0002] Several methods and approaches for fault location in high voltagepower systems have been developed and tried. One approach has been touse voltage/current transducers located at terminals, between which thepower lines to be monitored run.

[0003] One such system is disclosed in U.S. Pat. No. 5,455,776 where thetransducers are connected to transducer blocks and a fault locationestimation processor. The system uses positive or negative sequencenetworks.

[0004] Iterative calculations are used in U.S. Pat. No. 5,455,776 fordetermining the synchronization angle. In the disclosed method of U.S.Pat. No. 5,455,776, the unknown synchronization angle (δ) is to becalculated by an iterative Newton-Raphson method and after this, thefault distance can be determined. The Newton-Raphson approach utilizedin the method, starts from the initial guess for the synchronizationangle set at a certain pre-defined value (usually equal to zero). As aresult of iterative computations the nearest mathematical solution(which is the closest to the assumed initial guess) is reached. Thisapproach seems to be reasonable for a majority of applications, as theother solution for the synchronization angle (which is mathematicallypossible but rejected here) is usually far away from the assumed initialguess and outside of the reasonable range. However, for some heavy faultcases (large difference of the detected instants of a fault occurrenceperformed at both line ends) there is a risk that the rejected solutionis regarded as a correct result, while the reached solution is false.

BRIEF DESCRIPTION OF THE INVENTION

[0005] The object of the present invention is to provide a method forlocation of a fault utilizing unsynchronized measurements of three phasevoltages and currents acquired at the line terminals withoutsynchronization. Phasors for symmetrical components of the measuredquantities are determined and used in the fault location algorithm.According to one embodiment, positive sequence phasors of post-faultquantities are used for estimation of the distance to fault and it isdistinctive that such an estimation of a distance to fault is performedwithout essentially involving iterative techniques. In this embodiment,the fault location algorithm step is performed regardless of the faulttype. In later steps, the type of fault may be obtained. According toanother embodiment of the invention, at the occurrence of a fault, thetype of fault is determined. If it is determined that it is not athree-phase balanced fault, negative sequence phasors are used for theestimation of the distance to fault. On the other hand, if it is athree-phase balanced fault, the incremental positive sequence phasorsare used. The incremental positive sequence phasors are to be understoodas the difference between the post-fault and the pre-fault values.

[0006] In addition, only for some—rather rare cases—the fault locationalgorithm directs into the two optional branches (A and B). This is donejust for selection of the valid result for a distance to fault.

[0007] The first branch (A) is based on comparison of the values of thedetermined synchronization angle. The pre-fault measurements (A1) orpost-fault measurements from the sound phases (A2) are utilized. Asunder three-phase balanced faults there are no sound phases, thus, whenusing the sub-branch (A2) for such faults, the impedance for the faultpath is determined by taking the positive sequence phasors. Theresistive character of this impedance indicates the valid solution, asit will be used also in the sub-branch (B1).

[0008] The other branch (B) requires distinguishing whether it is athree-phase balanced fault or the other unsymmetrical fault type. Forthree-phase balanced faults (B1) the impedance for the fault path isdetermined by taking the positive sequence phasors and the resistivecharacter of this impedance indicates the valid solution. For the othertypes of fault (B2), the negative sequence quantities are used forestimation of a distance to fault.

[0009] In order to provide high accuracy of fault location the estimateinitially obtained for the distance to fault undergoes refining bycompensating for shunt capacitances of a line.

[0010] Since the zero sequence quantities are not involved in thepresented algorithm the algorithm is applicable for locating faults inboth a single line and parallel mutually coupled transmission lines.

[0011] The method according to the present invention differssubstantially from the method introduced in U.S. Pat. No. 5,455,776. Thesynchronization angle (δ), which relates the measurements from both lineterminals to the common time base, is not involved here in the faultdistance calculation itself. In fact, the synchronization angle is usedin the presented fault location algorithm. However, it is used for theother purpose, namely for selecting the valid result for a distance tofault and is optional and not obligatory. In the algorithm according tothe present invention the optional way of the selection, as A (A1 or A2)can be replaced by the way B (B1 together with B2), where the latterdoes not involve the synchronization angle.

[0012] The proposed method avoids the iterative calculations during thedetermination of the fault distance and at the same time considers allthe mathematically possible solutions for the sought fault distance andin consequence of the synchronization angle. This guaranties thatirrespective of parameters of the considered transmission network andthe fault specifications the unique solution is always achieved. Thus,as for example, the algorithm can be adapted for locating faults inseries-compensated lines, which are considered as extremely complexnetworks. Providing the unique solution is thus especially desired forsuch applications.

[0013] According to one embodiment, the fault location procedure startswith solving the quadratic equation involving positive sequence phasorsonly. This gives two solutions for the fault distance and only one ofthem corresponds to the actual value. The valid result in vast majorityof the cases is directly obtained if only a single solution for thefault distance lies within a line length. While, in some rather veryrare cases further selection of the valid result is required and noiterative calculations are applied for that too.

[0014] In case of three-phase symmetrical faults the original selectionprocedure is applied. This procedure selects the valid result bychecking which the solution results in a lower imaginary part (ideallyought to be zero) of the estimated impedance for a fault path.

[0015] According to another embodiment, the first step is to determinethe type of fault. After that, depending on the type of fault, eitherthe quadratic equation involving negative sequence phasors is solved, orthe quadratic equation involving incremental positive sequence phasorsis solved.

[0016] According to an optional embodiment of the invention, improvedaccuracy of fault location may be obtained by introducing a calculationin which the shunt capacitances of a line are compensated for. Thisinvolves iterative calculations, however limited to a simple iterationcalculation, in which a total of two iterations usually provides highaccuracy. The compensation of shunt capacitances is an optionalrefinement of the algorithm and is performed at a late stage.

[0017] The present invention is applicable to transmission networks upto and over 400 kV as well as to distribution networks.

[0018] These and other aspects of the invention and its benefits willappear from the detailed description of the invention and from theaccompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

[0019] In the following detailed description of the invention, referencewill be made to the accompanying drawings, of which

[0020]FIG. 1a shows equivalent diagram of a two terminal line for thepositive sequence,

[0021]FIG. 1b shows equivalent diagram of a two terminal line for thenegative sequence,

[0022]FIG. 2 shows the Π model of a line for the pre-fault positivesequence with including the shunt capacitances,

[0023]FIG. 3 shows determination of the positive sequence phasors forthe pre-fault phase currents and voltages acquired at the substations Aand B,

[0024]FIG. 4 shows an equivalent circuit diagram for a transmission lineaffected by a three-phase symmetrical fault,

[0025]FIG. 5 shows an equivalent circuit diagram (positive sequence) forthe first iteration of the compensation for shunt capacitances of aline,

[0026]FIG. 6 shows a flow-chart of one example of a fault locationalgorithm according to the present invention,

[0027]FIG. 7 shows a flow-chart of another example of a fault locationalgorithm according to the present invention, and

[0028]FIG. 8 shows an example of a device and system for carrying outthe method.

DETAILED DESCRIPTION OF THE INVENTION Characteristic of theUnsynchronized Measurements at the Line Terminals

[0029] It is considered that the measurements at the line terminals areperformed without any synchronization. Thus, a fault occurrence instantis the only time relation between the measurement data from the lineterminals. Physically, the actual fault occurrence instant (say, t₀) isthe same for both ends data. However, in real life a fault can bedetected by the fault detectors (contained in protective relays or faultlocators at particular line ends) in certain instants (at the substationA: t_(A) and at the substation B: t_(B)), which do not correspond to theactual inception instant (t₀). So, taking into account that in general:

t_(A)≠t₀

t_(B)≠t₀

t_(A)≠t_(B)

[0030] one can expect that at both the ends the measurements fromslightly shifted intervals can be frozen and used as the input data forthe fault location algorithm. In some heavy cases (far end faults withhigh fault resistance) fault detection can be delayed even by fewsampling intervals T_(s) [say (3÷4)T_(s)], thus, for a typical samplingfrequency equal to 1000 Hz, this corresponds to the angle: [(54÷72)deg].At the same time at the opposite line terminal there could be no delayin fault detection at all or it could be very small—for example equal toa single T_(s). Thus, the shift of the frozen intervals of themeasurements at both the ends can correspond even to a few samplingintervals T_(s). Moreover, this shift does not have to be equal to themultiple of the sampling period T_(s) as, in general, the samplinginstants at both the line ends do not coincide due to free runningclocks which control the sampling at the line terminals.

[0031] The mentioned shift in time domain of the acquired samples ofvoltages and currents at the line terminals corresponds to thesynchronization angle (δ) when the phasors of the measured quantitiesare considered. The synchronization angle is to some extent a randomquantity and only the possible range for it, located around zero (zerois for ideal synchronization), can be defined (assessed) for particularapplication. Thus, the synchronization angle is treated as the extraunknown of the fault location algorithm.

BASICS OF THE FAULT LOCATION ALGORITHM

[0032]FIG. 1. presents equivalent diagrams of a single line for thepositive (FIG. 1a) and for the negative sequence (FIG. 1b) quantities.

[0033] Impedance of a line for the negative sequence (Z _(L2)) isassumed in all further considerations (as in reality) as equal to theimpedance for the positive sequence (Z _(L1)):

Z _(L2)=Z _(L1)  (1)

[0034] Moreover, all the phasors in FIG. 1a, b are considered as relatedto the time basis of the phasors measured at the substation B (V _(B1),I _(B1), V _(B2), I _(B2)), which are taken here as a reference. Themeasurements from the substations A and B are not synchronous and thusthe measurements performed at the substation A are synchronized“artificially” to the measurements performed at the substation B, whichare taken here as a reference. For this purpose the synchronizationphase shift term (e^(jδ)), where: δ—unknown synchronization angle, isintroduced. The synchronization phase shift term is included for both,the voltage and the current phasors from the substation A (V_(A1)e^(jδ), I _(A1)e^(jδ), V _(A2)e^(jδ), I _(A2)e^(jδ)).

[0035] The equivalent circuit diagram for the positive sequencequantities (FIG. 1a) can be described with the following two equations:

V _(A1) e ^(jδ) −dZ _(L1) I _(A1) e ^(jδ) −V _(F1)=0  (2)

V _(B1)−(1−d) Z _(L1) I _(B1) −V _(F1)=0  (3)

[0036] where:

[0037]V _(A1), V _(B1)—phasors of the positive sequence voltagesmeasured at the substations A and B, respectively,

[0038]I _(A1), I _(B1)—phasors of the positive sequence currentsmeasured at the substations A and B, respectively,

[0039]V _(F1)—unknown phasor of the positive sequence component of thevoltage drop across the fault path,

[0040]Z _(L1)—impedance of the whole line for the positive sequence,

[0041] d—unknown distance to fault [pu], counted from the substation A,

[0042] δ—unknown synchronization angle.

[0043] By subtracting equations (2) and (3) one obtains equation (4) inwhich the unknown quantity V _(F1) is eliminated and there are twounknowns-fault distance (d) and synchronization angle (δ):

V _(A1) e ^(jδ) −V _(B1) +Z _(L1) I _(B1) −dZ _(L1)( I _(A1) e ^(jδ) +I_(B1))=0  (4)

[0044] The obtained equation (4) is considered for the phasors and thuscan be resolved into its real and imaginary parts. In this way both theunknowns (d and δ) can be found.

[0045] The new approach according to the method of the present inventionfor solving equation (4), is to transfer it into the quadratic algebraicequation for the unknown fault distance (d). For this purpose onedetermines the term exp(jδ) expressed by the formula: $\begin{matrix}{^{j\quad \delta} = \frac{{\underset{\_}{V}}_{B1} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} + {d{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}}}{{\underset{\_}{V}}_{A1} - {d{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}}}} & (5)\end{matrix}$

[0046] Calculation of absolute values for both the sides of (5) gives:

A ₂ d ² +A ₁ d+A ₀=0  (6)

[0047] where:

[0048] d—unknown fault distance,

[0049] A₂, A₁, A₀—coefficients (real numbers) expressed duly with theunsynchronized measurements of the positive sequence phasors of voltages(V _(A1), V _(B1)) and currents (I _(A1), I _(B1)) at the line terminalsand with the positive sequence impedance of a line (Z _(L1)):

A ₂ =|Z _(L1) I _(A1)|² −|Z _(L1) I _(B1)|²

A ₁=−2real{ V _(A1)( Z _(L1) I _(A1))*+( V _(B1) −Z _(L1) I _(B1))( Z_(L1) I _(B1))*}

A ₀ =V _(A1)|² −|V _(B1) −Z _(L1) I _(B1)|²  (6a)

[0050]X* denotes conjugate of X, |X| denotes absolute value of X.

[0051] Complete derivation of equations (6)-(6a) is provided in APPENDIX1.

[0052] Solution of the quadratic equation (6) with the coefficientsdefined in (6a) gives two solutions for a distance to fault (d₁, d₂):$\begin{matrix}{{d_{1} = \frac{{- A_{1}} - \sqrt{A_{1}^{2} - {4A_{2}A_{0}}}}{2A_{2}}}{d_{2} = \frac{{- A_{1}} + \sqrt{A_{1}^{2} - {4A_{2}A_{0}}}}{2A_{2}}}} & \left( {6b} \right)\end{matrix}$

[0053] Flow-chart of the complete fault location algorithm is presentedin FIG. 6. According to this flow-chart the further steps are asfollows.

[0054] If only one solution according to equation (6b) is within a linelength:

0<(d₁ or d₂)<1 pu  (6c)

[0055] then the value satisfying equation (6c) is taken as the validresult (d_(ν)) while the other solution (indicating a fault as outside aline) is rejected.

[0056] In contrast, if both the results for a fault distance accordingto equation (6b) indicate a fault as in a line:

0<(d₁ and d₂)<1 pu (6d)

[0057] then the selection of the valid result is required to beperformed.

[0058] The selection of the valid result is performed as follows. First,the values of the synchronization angle corresponding to both thesolutions of equation (6b) are determined: $\begin{matrix}{^{j\quad \delta_{1}} = \frac{{\underset{\_}{V}}_{B1} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} + {d_{1}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}}}{{\underset{\_}{V}}_{A1} - {d_{1}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}}}} & (7) \\{^{j\quad \delta_{2}} = \frac{{\underset{\_}{V}}_{B1} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} + {d_{2}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}}}{{\underset{\_}{V}}_{A1} - {d_{2}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}}}} & (8)\end{matrix}$

[0059] Generally, these values (δ₁ and δ₂) could lie in the wholemathematical range, which is considered when trigonometric relations areutilized—the range: (−π÷π). However, for particular application thisrange can be considered as somehow reasonably shortened—to the certainrange, which can be expected: (−δ_(short)÷δ_(short)). For example,assuming for this range:

(−δshort÷δ_(short))=(−π/2÷π/2)  (8a)

[0060] will provide even certain safety margin. This is so, as the valueof the angle, assumed in equation (8a) as equal to (π/2) is high enough.It corresponds to 4 sampling periods for a difference in fault detectioninstants at the line ends plus a single sampling period for having nosynchronized clocks controlling sampling at both the line terminals (at1000 Hz sampling frequency).

[0061] If only one value of the synchronization angle lies in theassumed range (−δ_(short)÷δ_(short)):

−δ_(short)<(δ₁ or δ₂)<δ_(short)  (8a)

[0062] then, this value of the synchronization angle (δ₁ or δ₂) whichsatisfies equation (8a) indicates a valid solution for a distance tofault. Thus, when only the value (δ₁) satisfies (8a), then the value ofa distance to fault (d₁) is taken as the valid solution (δ_(ν)=d₁).Similarly, if the value (δ₂) satisfies (8a) then the value of a distanceto fault (d₂) is taken as the valid solution (δ_(ν)=d₂).

[0063] In contrast, if both the values of the synchronization angle liein the assumed range (−δ_(short)÷δ_(short)), i.e.:

δ_(short)<(δ₁ and δ₂)<δ_(short)  (8b)

[0064] further selection is required to be performed.

[0065] Further selection, if (8b) is satisfied, can be performed in thefollowing two optional ways (A and B).

A: Comparison of the Calculated Values of the Synchronization Angles

[0066] A1: Comparison of the values of the synchronization angles (7),(8) corresponding to both the solutions (6b) with the value of the angle(δ_(m)), which is determined with the pre-fault measurements of currentsand voltages for the positive sequence.

[0067] The values for the shunt branches in the model of FIG. 2 aredefined as follows: B ₁=jω₁C₁, where: C₁ is the positive sequence shuntcapacitance of the whole line.

[0068] In order to determine the value of the synchronization angle(δ_(m)) the computation starts from calculating positive sequencephasors of the pre-fault phase voltages and currents acquired at thesubstations A and B (FIG. 3). For example, taking the pre-fault currentsfrom phases (a, b, c) at the station A (I _(A) _(—) _(pre) _(—) _(a), I_(A) _(—) _(pre) _(—) _(b), I _(A) _(—) _(pre) _(—) _(c)) the positivesequence phasor (I _(A) _(—) _(pre) _(—) ₁) is calculated. Analogously,the positive sequence phasor (V _(A) _(—) _(pre) _(—) ₁) is calculatedfrom the pre-fault phase voltages from the station A. The sameprocessing is for the pre-fault phase currents and voltages from thesubstation B (FIG. 3).

[0069] The value of the synchronization angle (δ_(m)) is calculated fromthe following condition for the circuit of FIG. 2:

I _(Ax) =−I _(Bx)  (A1_(—)1)

where:

I _(Ax) =I _(A) _(—) _(pre) _(—) ₁ e ^(jδ) ^(_(m)) −j0.5ω₁ C ₁ V _(A)_(—) _(pre) _(—) ₁ e ^(jδ) ^(_(m))

I _(Bx) =I _(B) _(—) _(pre) _(—) ₁ −j0.5ω₁ C ₁ V _(B) _(—) _(pre) _(—) ₁

[0070] After some rearrangements, the formula (A1_(—)1) leads to thefollowing formula for the synchronization angle (δ_(m)): $\begin{matrix}{\delta_{m} = {{{angle}\left( {P + {jQ}} \right)} = {\tan^{- 1}\left( \frac{{N_{2}D_{1}} - {N_{1}D_{2}}}{{N_{1}D_{1}} + {N_{2}D_{2}}} \right)}}} & \left( {{A1\_}9} \right)\end{matrix}$

[0071] where:

N ₁=−real( I _(B) _(—) _(pre) _(—) ₁)−0.5ω₁ C ₁imag( V _(B) _(—) _(pre)_(—) ₁)

N ₂=−imag( I _(B) _(—) _(pre) _(—) ₁)+0.5ω₁ C ₁real( V _(B) _(—) _(pre)_(—) ₁)

D ₁=real( I _(A) _(—) _(pre) _(—) ₁)+0.5ω₁ C ₁imag( V _(A) _(—) _(pre)_(—) ₁)

D ₂=imag( I _(A) _(—) _(pre) _(—) ₁)−0.5ω₁ C ₁real( V _(A) _(—) _(pre)_(—) ₁)

[0072] Complete derivation of (A1_(—)9) is provided in APPENDIX 2.

[0073] The value of the synchronization angle δ_(m) (calculated inequation (A1_(—)9)) is compared with the values of the synchronizationangle (δ₁, δ₂), introduced in equations (7)-(8) and calculated:$\begin{matrix}\begin{matrix}{\delta_{1} = {{angle}\left( \frac{{\underset{\_}{V}}_{B1} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} + {d_{1}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}}}{{\underset{\_}{V}}_{A1} - {d_{1}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}}} \right)}} \\{= {{angle}\left( {P_{1} + {jQ}_{1}} \right)}} \\{= {\tan^{- 1}\left( \frac{Q_{1}}{P_{1}} \right)}}\end{matrix} & \left( {{A1\_}10} \right)\end{matrix}$

[0074] where: $\begin{matrix}{{P_{1} = {{real}\left( \frac{{\underset{\_}{V}}_{B1} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} + {d_{1}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}}}{{\underset{\_}{V}}_{A1} - {d_{1}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}}} \right)}}{Q_{1} = {{imag}\left( \frac{{\underset{\_}{V}}_{B1} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} + {d_{1}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}}}{{\underset{\_}{V}}_{A1} - {d_{1}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}}} \right)}}{and}\begin{matrix}{\delta_{2} = {{angle}\left( \frac{{\underset{\_}{V}}_{B1} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} + {d_{2}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}}}{{\underset{\_}{V}}_{A1} - {d_{2}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}}} \right)}} \\{= {{angle}\left( {P_{2} + {jQ}_{2}} \right)}} \\{= {\tan^{- 1}\left( \frac{Q_{2}}{P_{2}} \right)}}\end{matrix}} & \left( {{A1\_}11} \right)\end{matrix}$

[0075] where:$P_{2} = {{real}\left( \frac{{\underset{\_}{V}}_{B1} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} + {d_{2}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}}}{{\underset{\_}{V}}_{A1} - {d_{2}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}}} \right)}$$Q_{2} = {{imag}\left( \frac{{\underset{\_}{V}}_{B1} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} + {d_{2}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}}}{{\underset{\_}{V}}_{A1} - {d_{2}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}}} \right)}$

[0076] The selection of the valid result (d_(ν)) from the solutions ofthe quadratic equation (6) for the fault distance [(d₁ or d₂)—eq. (6b)]is performed as follows:

if |δ₂−δ_(m)|>|δ₁−δ_(m)| then d _(ν) =d ₁

else

d_(ν)=d₂  (A1_(—)12)

[0077] end

[0078] A2: Comparison of the values of the synchronization angles(A1_(—)10)-(A1_(—)11), corresponding to both the solutions (6b) with thevalue of the angle (δ_(sound)), which is determined with the post-faultmeasurements of currents and voltages taken from the sound phase.

[0079] The procedure is analogous as in the section A1 but instead ofthe positive sequence phasors of the pre-fault quantities the phasorsfor the particular sound phase voltage and current are utilized.

[0080] First, it ought to be checked whether there is any sound phase.For three phase balanced faults there is no sound phase and for such acase the procedure based on checking the imaginary part of the estimatedfault path impedance is utilized (this procedure will be considered inthe next parts of this document—the branch B1 in the flow-chart (FIG.6)).

[0081] For example if the a-g fault occurs then the quantities from thephase b or from the phase c can be utilized. Taking the phase quantitiesfrom a particular sound phase the value of the synchronization angle(δ_(sound)) is determined (analogously as it is shown for determiningthe value of the synchronization angle δ_(m)—equation (A1_(—)9)).

[0082] The selection of the valid result (d_(ν)) from the solutions ofthe quadratic equation (6) for the fault distance (d₁ or d₂), calculatedin (6b), is performed as follows:

if |δ₂−δ_(sound)|>|δ₁−δ_(sound)| then d _(ν) =d ₁

else

d_(ν)=d₂  (A2_(—)1)

[0083] end

[0084] where:

[0085] δ₁, δ₂—the values of the synchronization angle calculated inequation (A1_(—)10)-(A1_(—)11).

B. Analytically, Without Considering the Synchronization Angle

[0086] B1: For three-phase balanced faults—by considering the impedanceof a fault path, estimated by taking the positive sequence measuredquantities. The values of the fault path impedance [Z _(F1)(d₁) and Z_(F2)(d₂)], corresponding to both the solutions of the equation (6) fordistance to a fault (d₁, d₂), are calculated. For real faults the faultpath impedance is resistive. Thus, this impedance [Z _(F1)(d₁) or Z_(F2)(d₂)], which has smaller imaginary part (ideally equal to zero),indicates the valid solution for a distance to fault (d₁ or d₂),calculated in (6b).

[0087] Definition of the fault path impedance under balanced three-phasefaults follows. FIG. 4 presents equivalent diagram used for that.

[0088] In the circuit of FIG. 4 (a three-phase fault on a line) thepositive sequence components of currents and voltages are the onlycomponents, which are present in the signals. For real faults the faultpath impedance is composed of a fault resistance only: Z _(F)=R_(F). Theselection procedure of the valid solution is based on checking which ofthe estimated fault path impedances [Z _(F1)(d₁) or Z _(F2)(d₂)] iscloser to the condition Z _(F)=R_(F).

[0089] Taking the left-hand side loop (LHS) in the circuit of FIG. 4 thefollowing voltage formula can be written down:

V _(A1) e ^(jδ) −dZ _(L1) I _(A1) e ^(jδ) −Z _(F)( I _(A1) e ^(jδ) +I_(B1))=0  (a)

[0090] The values of the fault path impedance are calculated as:

Z _(F1)(d ₁)= P ₂ d ₁ ² +P ₁ d ₁ +P ₀  (9)

Z _(F2)(d ₂)= P ₂ d ₂ ² +P ₁ d ₂ +P ₀  (10)

[0091] where:${\underset{\_}{P}}_{2} = \frac{{- {\underset{\_}{Z}}_{L1}^{2}}{\underset{\_}{I}}_{A1}{\underset{\_}{I}}_{B1}}{{{\underset{\_}{I}}_{A1}{\underset{\_}{V}}_{B1}} + {{\underset{\_}{I}}_{B1}{\underset{\_}{V}}_{A1}} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}{\underset{\_}{I}}_{B1}}}$${\underset{\_}{P}}_{1} = \frac{{{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}{\underset{\_}{V}}_{A1}} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}{\underset{\_}{V}}_{B1}} + {{\underset{\_}{Z}}_{L1}^{2}{\underset{\_}{I}}_{A1}{\underset{\_}{V}}_{B1}}}{{{\underset{\_}{I}}_{A1}{\underset{\_}{V}}_{B1}} + {{\underset{\_}{I}}_{B1}{\underset{\_}{V}}_{A1}} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}{\underset{\_}{I}}_{B1}}}$${\underset{\_}{P}}_{0} = \frac{{{\underset{\_}{V}}_{A1}{\underset{\_}{V}}_{B1}} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}{\underset{\_}{V}}_{A1}}}{{{\underset{\_}{I}}_{A1}{\underset{\_}{V}}_{B1}} + {{\underset{\_}{I}}_{B1}{\underset{\_}{V}}_{A1}} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}{\underset{\_}{I}}_{B1}}}$

[0092] Complete derivation of equations (9)-(10) is provided in APPENDIX3.

[0093] The impedance (equations (9) or (10)), that has smaller imaginarypart (ideally equal to zero), indicates the valid solution. Theselection of the valid result (d_(ν)) from the solutions of thequadratic equation (6) for the fault distance (d₁ or d₂) is performed asfollows:

if |imag( Z _(F1)(d ₁))|<|imag( Z _(F2)(d ₂))| then d _(ν) =d ₁

else

d_(ν)=d₂  (11)

[0094] end

[0095] Note: The criterion for selection of the valid result based onchecking the imaginary part of the fault path impedance, estimated withuse of the positive sequence post-fault quantities (11) is novel.

[0096] B2: For the other fault types (all faults except the balancedthree-phase faults) the quadratic equation stated for negative sequencequantities is utilized.

[0097] With Reference to FIG. 1b (equivalent circuit diagram for thenegative sequence) and proceeding analogously, as it has been done forderiving the quadratic equation for the positive sequence (6), oneobtains the quadratic equation for the sought fault distance (d) as:

B ₂ d ² +B ₁ d+B ₀=0  (12)

[0098] where: B₂, B₁, B₀ are the real number coefficients expressedanalogously as the coefficients of (6), but with use of the negativesequence quantities:

B ₂ =|Z _(L1) I _(A2)|² −|Z _(L1) I _(B2)|²

B ₁=−2real{ V _(A2)( Z _(L1) I _(A2))*+( V _(B2) −Z _(L1) I _(B2))( Z_(L1) I _(B2))*}

B ₀ =V _(A2)|² −|V _(B2) −Z _(L1) I _(B2)|²  (12a)

[0099] By solving equation (12), taking into account equation (12a), thenext two solutions for the fault distance (d₃, d₄) are obtained:$\begin{matrix}{{d_{3} = \frac{{- B_{1}} - \sqrt{B_{1}^{2} - {4B_{2}B_{0}}}}{2B_{2}}}{d_{4} = \frac{{- B_{1}} + \sqrt{B_{1}^{2} - {4B_{2}B_{0}}}}{2B_{2}}}} & \left( {12b} \right)\end{matrix}$

[0100] The solutions, taken out of all four solutions (d₁, d₂, d₃, d₄),which coincide (d_(i)=d_(j), where: i=1 or 2, j=3 or 4) indicate thevalid solution for the fault distance (d_(ν)). In practice, there aresome errors in fault distance estimation (as for example, shuntcapacitances of a line are not taken into account at this stage of thefault location algorithm) and instead of the ideal condition d_(i)=d_(j)the following can be applied:

|d _(i) −d _(j)|=min  (13)

[0101] where: i=1 or 2, j=3 or 4

[0102] As a result of the selection criterion (13) the valid result isobtained as:

d_(ν)=d_(j)  (13a)

[0103] Note: The solution (d_(j))—obtained by solving the quadraticequation for the negative sequence quantities, and not the solution(d_(i))—obtained by solving the quadratic equation for the positivesequence quantities, is taken as the valid result (13a). This is so, asneglecting shunt capacitances of a line influences stronger the accuracyof fault location (there are larger errors), when performed with thepositive sequence quantities than with the negative sequence quantities.

[0104] According to an optional embodiment the fault location accuracymay be refined by finally compensating for shunt capacitances of a line.The compensation procedure is iterative and ends if the consecutiveresults for the distance to a fault differ by less than the set marginfor the convergence.

[0105] First iteration of the compensation (subscript: comp_(—)1) isdescribed below.

[0106] In the compensation procedure the phasors of currents arecompensated for line shunt capacitances while the original phasors forthe voltage signals are applied. Referring to FIG. 5, the compensationfor the positive sequence currents is performed according to:

I _(A1) _(—) _(comp) _(—) ₁ =I _(A1) −j0.5ω₁ C ₁ d _(ν) V _(A1)  (14)

I _(B1) _(—) _(comp) _(—) ₁ =I _(B1) −j0.5ω₁ C ₁(1−d _(ν)) V _(B1)  (15)

[0107] where:

[0108] d_(ν)—fault distance (the valid result) obtained without takinginto account the shunt capacitances of a line,

[0109] C₁—shunt capacitance of a whole line.

[0110] Taking the phasors compensated for shunt capacitances ((14)-(15))the quadratic algebraic equation for a fault distance before thecompensation (6) transforms to the following quadratic algebraicequation for the improved fault distance:

A ₂ _(—) _(comp) _(—) ₁ d ² _(comp) _(—) ₁ +A ₁ _(—) _(comp) _(—) ₁ d_(comp) _(—) ₁ +A ₀ _(—) _(comp) _(—) ₁=0  (16)

[0111] where:

[0112] d_(comp) _(—) ₁—the improved fault distance result (result of thefirst iteration of the compensation),

[0113] A₂ _(—) _(comp) _(—) ₁, A₁ _(—) _(comp) _(—) ₁, A₀ _(—) _(comp)_(—) ₁—coefficients (real numbers) expressed duly with theunsynchronized measurements at the line terminals and with the positivesequence impedance of a line (Z _(L1)). The following measurements areused: the original positive sequence phasors of voltages (V _(A1), V_(B1)) and the phasors of currents compensated for a line shuntcapacitances (I _(A1) _(—) _(comp) _(—) ₁, I _(B1) _(—) _(comp) _(—) ₁):

A ₂ _(—) _(comp) _(—) ₁ =|Z _(L1) I _(A1) _(—) _(comp) _(—) ₁|² −|Z_(L1) I _(B1) _(—) _(comp) _(—) ₁|²

A ₁ _(—) _(comp) _(—) ₁=−2real{ V _(A1)( Z _(L1) I _(A1) _(—) _(comp)_(—) ₁)*+( V _(B1) −Z _(L1) I _(B1) _(—) _(comp) _(—) ₁)( Z _(L1) I_(B1) _(—) _(comp) _(—) ₁)*}

A ₀ _(—) _(comp) _(—) ₁ =|V _(A1)|² −|V _(B1) −Z _(L1) I _(B1) _(—)_(comp) _(—) ₁|²  (16a)

[0114] By solving equation (16), and taking (16a), the two solutions forthe fault distance (d_(comp) _(—) _(1A), d_(comp) _(—) _(1B)) areobtained: $\begin{matrix}{d_{{comp\_}1A} = \frac{{- A_{1{\_ comp}\_ 1}} - \sqrt{A_{1{\_ comp}\_ 1}^{2} - {4A_{2{\_ comp}\_ 1}A_{0{\_ comp}\_ 1}}}}{2A_{2{\_ comp}\_ 1}}} & \left( {16b} \right) \\{d_{{comp\_}1B} = \frac{{- A_{1{\_ comp}\_ 1}} + \sqrt{A_{1{\_ comp}\_ 1}^{2} - {4A_{2{\_ comp}\_ 1}A_{0{\_ comp}\_ 1}}}}{2A_{2{\_ comp}\_ 1}}} & \left( {16c} \right)\end{matrix}$

[0115] As a result of the first iteration of the compensation theimproved value for the fault distance is obtained (d_(ν) _(—) _(comp)_(—) ₁). Selection of one particular solution from equations (16b) or(16c), is straightforward. If before the compensation the solution d₁was selected as the valid result (d_(ν)=d₁), then we take d_(comp) _(—)_(1A) as the valid result after the first iteration of the compensation(d_(ν) _(—) _(comp) _(—) ₁=d_(comp) _(—) _(1A)). Similarly, if, beforethe compensation the solution d₂ was selected as the valid result(d_(ν)=d₂) then we take d_(comp) _(—) _(1B) as the valid result afterthe first iteration of the compensation (d_(ν) _(—) _(comp) _(—)₁=d_(comp) _(—) _(1B)).

[0116] Next iterations of the compensation are performed analogously.The value of a distance to fault calculated in a particular iteration(say n^(th) iteration): (d_(ν) _(—) _(comp) _(—) _(n)) is taken forcompensation of the phasors of the currents for shunt capacitances (likein equations (14)-(15)) and after to be introduced in the next iteration((n+1)^(th) iteration).

[0117] The iterations are continued until the convergence with thepre-defined margin (d_(margin)) is achieved:

|d _(ν) _(—) _(comp) _(—) _(i) −d _(ν) _(—) _(comp) _(—) _(i−1) |<d_(margin)  (17)

[0118] where:

[0119] the index (i) stands for the present iteration while the index(i−1) for the preceding iteration.

[0120] Solving the quadratic algebraic equation (6) for the unknownfault distance (d) is the first step of the sequence of computations inthe fault location algorithm presented in FIG. 6. This step is performedregardless the fault type. Information on the fault type—in terms:whether it is the three-phase balanced fault or any other fault—could berequired in the next steps. An alternative way of computations, as shownin FIG. 7, is based on utilizing the above mentioned information on thefault type (whether it is the three-phase balanced fault or any otherfault) at the very beginning of the calculation algorithm.

[0121] In the alternative algorithm, the calculation algorithm branchesinto two different paths, which are used depending upon what type of afault has been recognized.

[0122] In case of all faults apart from three-phase balanced faults, theunknown fault distance is calculated by solving the quadratic algebraicequation (12). This gives two solutions for a distance to fault (d₃, d₄)as in equation (12b). If only one solution according to equation (12b)is within a line length then this value is taken as the valid result(d_(ν)). Otherwise (both solutions from equation (12b) are within a linelength), the selection of the valid solution (d₃ or d₄) is required.This selection can be performed in two optional ways:

[0123] The first way of the selection (A) is based on determining thevalues of the synchronization angle, as it is defined in equations (7)and (8), but in this case the values correspond to the solutions (d₃,d₄). Firstly, satisfying of the condition of equation (8a) has to bechecked. If equation (8a) is satisfied, then the solution that satisfiesequation (8a) is taken as the valid. In contrast (when equation (8b) issatisfied), the other two optional procedures of the selection,described in the sections A1 and A2 above, have to be applied.

[0124] The second way of the selection (B) is based on solving thequadratic algebraic equation for the positive sequence quantities (6).This gives two solutions for a distance to fault (d₁, d₂), which arecalculated according to equation (6b). The solutions, taken out of allfour solutions (d₁, d₂, d₃, d₄), which coincide (d_(i)=d_(j), where: i=1or 2, j=3 or 4) indicate the valid solution for the fault distance(d_(ν)). This is performed by using equations (13) and (13a).

[0125] In case of three-phase balanced faults, the unknown faultdistance is calculated by solving the quadratic algebraic equation,which is formulated for the incremental positive sequence quantities.The particular incremental positive sequence component is understood asthe difference between the post-fault and the pre-fault values.

[0126] Proceeding analogously, as it has been done for deriving equation(6), derives the quadratic equation for the incremental positivesequence quantities. The positive sequence quantities from equation (6)are replaced by the corresponding incremental positive sequencequantities. Solution of such obtained quadratic algebraic equationprovides two solutions for a distance to fault (d₅, d₆), calculatedanalogously as in equation (6b). If only one solution (d₅ or d₆) iswithin a line then this solution is taken as the valid solution (d_(ν)).Otherwise (both solutions (d₅ and d₆) are within a line length), theselection of the valid solution (d₅ or d₆) is required. This selectioncan be performed in the analogous ways, as applied for the case of allfaults, but with exception of three-phase symmetrical faults. Thereplacing of the solutions d₃, d₄ by the solutions d₅, d₆ is requiredfor that.

[0127] According to the embodiment the fault location accuracy may berefined by finally compensating for shunt capacitances of a line. Thecompensation procedure is as described by equations (14)-(17).

[0128]FIG. 7 shows an embodiment of a device for determining thedistance from a station, at one end of a transmission line, until theoccurrence of a fault on the transmission line according to thedescribed method, comprising certain measuring devices, measurementvalue converters, members for treatment of the calculating algorithms ofthe method, indicating means for the calculated distance to fault and aprinter for printout of the calculated fault.

[0129] In the embodiment shown, measuring devices 1 and 2 for continuousmeasurement of all the phase currents from the faulty line and phasevoltages are arranged in both stations A and B. In the measurementconverters 3 and 4, a number of these consecutively measured values,which in case of a fault are passed to a calculating unit 5, arefiltered and stored. The calculating unit is provided with thecalculating algorithms described, programmed for the processes neededfor calculating the distance to fault and the fault resistance. Thecalculating unit is also provided with known values such as theimpedance of the line. In connection to the occurrence of a faultinformation regarding the type of fault may be supplied to thecalculating unit for choosing the right selection path. When thecalculating unit has determined the distance to fault, it is displayedon the device and/or sent to remotely located display means. A printoutof the result may also be provided. In addition to signalling the faultdistance, the device can produce reports, in which are recorded measuredvalues of the currents of both lines, voltages, type of fault and otherassociated with a given fault at a distance.

[0130] The information in the form of a result for d_(ν) or d_(ν-comp)from the fault location system may also be embodied as a data signal forcommunication via a network to provide a basis for a control action. Thedistance d_(ν) or d_(ν-comp) may be sent as a signal for a controlaction such as: automatic notification to operational network centres offault and its location or to automatically start calculations todetermine journey time to location, which repair crew shall bedispatched to site, possible time taken to execute a repair, calculatewhich vehicles or crew member may be needed, how many shifts work percrew will be required and the like actions.

[0131] The calculating unit may comprise filters for filtering thesignals, A/D-converters for converting and sampling the signals and amicro processor. The micro processor comprises a central processing unitCPU performing the following functions: collection of measured values,processing of measured values, calculation of distance to fault andoutput of result from calculation. The micro processor further comprisesa data memory and a program memory.

[0132] A computer program for carrying out the method according to thepresent invention is stored in the program memory. It is to beunderstood that the computer program may also be run on general purposecomputer instead of a specially adapted computer.

[0133] The software includes computer program code elements or softwarecode portions that make the computer perform the said method using theequations, algorithms, data and calculations previously described. Apart of the program may be stored in a processor as above, but also in aRAM, ROM, PROM or EPROM chip or similar, and may also be run in adistributed fashion. The program in part or in whole may also be storedon, or in, other suitable computer readable medium such as a magneticdisk, CD-ROM or DVD disk, hard disk, magneto-optical memory storagemeans, in volatile memory, in flash memory, as firmware, or stored on adata server.

[0134] It is to be noted that the embodiment of the invention describedand shown in the drawings is to be regarded as a non-limiting example ofthe invention and that the scope of protection is defined by the patentclaims.

[0135] APPENDIX 1—Derivation Of the Quadratic Equation for PositiveSequence Quantities [Formulas: (6)-(6a)]

[0136] By computing absolute values of both the sides of (5) oneobtains: $\begin{matrix}{1 = \frac{{{\underset{\_}{V}}_{B1} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} + {d\quad {\underset{\_}{Z}}_{L1}} - {\underset{\_}{I}}_{B1}}}{{{\underset{\_}{V}}_{A1} - {d\quad {\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}}}}} & \left( {5a} \right)\end{matrix}$

[0137] or:

| V _(B1) −Z _(L1) I _(B1) +dZ _(L1) I _(B1) |=|V _(A1) −dZ _(L1) I_(A1)|  (5b)

[0138] Left-hand side of equation (5b) can be written down as:

L=|V _(B1) −Z _(L1) I _(B1) +dZ _(L1) I _(B1)|=|( V _(B1))_(real) +j( V_(B1))_(imag)−( Z _(L1) I _(B1))_(real) −j( Z _(L1) I _(B1))_(imag)++(dZ_(L1) I _(B1))_(real) +j(dZ _(L1) I _(B1))_(imag)|  (5c)

[0139] Continuing determination of the left-hand side of (5b) oneobtains: $\begin{matrix}\begin{matrix}{L = {{{\underset{\_}{V}}_{B1} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} + {d\quad {\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}}}}} \\{= \sqrt{\left( {\left( {\underset{\_}{V}}_{B1} \right)_{real} - \left( {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} \right)_{real} + \left( {d{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} \right)_{real}} \right)^{2} + \left( {\left( {\underset{\_}{V}}_{B1} \right)_{imag} - \left( {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} \right)_{imag} + \left( {d{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} \right)_{imag}} \right)^{2}}}\end{matrix} & \left( {5d} \right)\end{matrix}$

[0140] After rearrangements one obtains:

L={square root}{square root over ((L ₁ +L ₂ d)²+(L ₃ +L ₄ d)²)}  (5e)

[0141] where:

L ₁=(V _(B1))_(real)−( Z _(L1) I _(B1))_(real)

L ₂=( Z _(L1) I _(B1))_(real)

L ₃=( V _(B1))_(imag)−( Z _(L1) I _(B1))_(imag)

L ₄=( Z _(L1) I _(B1))_(imag)

[0142] Formula (5e) can also be written as:

L={square root}{square root over (L₅d²+L₆d+L₇)}  (5f)

[0143] where: $\begin{matrix}{L_{5} = {{L_{2}^{2} + L_{4}^{2}} = {\left\{ \left( {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} \right)_{real} \right\}^{2} + \left\{ \left( {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} \right)_{imag} \right\}^{2}}}} \\{L_{6} = {{2L_{1}L_{2}} + {2L_{3}L_{4}}}} \\{= {{2\left\{ {\left( {\underset{\_}{V}}_{B1} \right)_{real} - \left( {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} \right)_{real}} \right\} \left\{ \left( {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} \right)_{real} \right\}} +}} \\{{2\left\{ {\left( {\underset{\_}{V}}_{B1} \right)_{imag} - \left( {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} \right)_{imag}} \right\} \left\{ \left( {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} \right)_{imag} \right\}}}\end{matrix}$

 L ₇ =L ₁ ² +L ₃ ²={( V _(B1))_(real)−( Z _(L1) I _(B1))_(real)}²+{( V_(B1))_(imag)−( Z _(L1) I _(B1))_(imag)}²

[0144] Right-hand side of equation (5b) can be written down as:

R=|V _(A1) −dZ _(L1) I _(A1)|=|( V _(A1))_(real) +j( V _(A1))_(imag)−(dZ_(L1) I _(A1))_(real) −j(dZ _(L1) I _(A1))_(imag)|  (5g)

[0145] Continuing determination of the right-hand side of (5g) oneobtains: $\begin{matrix}\begin{matrix}{R = {{{\underset{\_}{V}}_{A1} - {d{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}}}}} \\{= \sqrt{\left\{ {\left( {\underset{\_}{V}}_{A1} \right)_{real} + \left( {{- d}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}} \right)_{real}} \right\}^{2} + \left\{ {\left( {\underset{\_}{V}}_{A1} \right)_{imag} + \left( {{- d}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}} \right)_{imag}} \right\}^{2}}}\end{matrix} & \left( {5h} \right)\end{matrix}$

[0146] After further rearrangements one obtains:

R={square root}{square root over ((R ₁ −R ₂ d)²+(R ₃ −R ₄ d)²)}  (5i)

[0147] where:

R ₁=( V _(A1))_(real)

R ₂=( Z _(L1) I _(A1))_(real)

R ₃=( V _(A1))_(imag)

R ₄=( Z _(L1) I _(A1))_(imag)

[0148] Formula (5i) can also be written as:

R={square root}{square root over (R₅d²+R₆d+R₇)}  (5j)

[0149] where:

R ₅ =R ₂ ² +R ₄ ²={( Z _(L1) I _(A1))_(real)}²+{( Z _(L1) I_(A1))_(imag)}²

[0150] $\begin{matrix}{R_{6} = {{{- 2}R_{1}R_{2}} - {2R_{3}R_{4}}}} \\{= {{{- 2}\left\{ \left( {\underset{\_}{V}}_{A1} \right)_{real} \right\} \left\{ \left( {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}} \right)_{real} \right\}} - {2\left\{ \left( {\underset{\_}{V}}_{A1} \right)_{imag} \right\} \left\{ \left( {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}} \right)_{imag} \right\}}}}\end{matrix}$

 R ₇ =R ₁ ² +R ₃ ²={( V _(A1))_(real)}²+{( V _(V1))_(imag)}²

[0151] Taking the results of the above derivations the formula (5b) canbe written down as:

{square root}{square root over (L ₅ d ² +L ₆ d+L ₇)}={squareroot}{square root over (R ₅ d ² +R ₆ d+R ₇)}  (5k)

[0152] where:

[0153] L₅, L₆, L₇—as in (5f),

[0154] R₅, R₆, R₇—as in (5j).

[0155] The formula (5k) results in the quadratic algebraic equation forthe unknown fault distance (d):

A ₂ d ² +A ₁ d+A ₀=0  (51)

[0156] where:

A ₂ =R ₅ −L ₅={( Z _(L1) I _(A1))_(real)}²+{( Z _(L1) I_(A1))_(imag)}²−{( Z 1 _(L1) I _(B1))_(real)}²−{( Z _(L1) I_(B1))_(imag)}²

A ₁ =R ₆ −L ₆=−2{( V _(A1))_(real)}{( Z _(L1) I _(A1))_(real)}−2{( V_(A1))_(imag)}{( Z _(L1) I _(A1))_(imag)}−2{( V _(B1))_(real)−( Z _(L1)I _(B1))_(real)}{( Z _(L1) I _(B1))_(real)}−2{( V _(B1))_(imag)−( Z_(L1) I _(B1))_(imag)}{( Z _(L1) I _(B1))_(imag)

[0157] $\begin{matrix}{A_{0} = {R_{7} - L_{7}}} \\{= {\left\{ \left( {\underset{\_}{V}}_{A1} \right)_{real} \right\}^{2} + \left\{ \left( {\underset{\_}{V}}_{A1} \right)_{imag} \right\}^{2} - \left\{ {\left( {\underset{\_}{V}}_{B1} \right)_{real} - \left( {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} \right)_{real}} \right\}^{2} -}} \\{\left\{ {\left( {\underset{\_}{V}}_{B1} \right)_{imag} - \left( {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} \right)_{imag}} \right\}^{2}}\end{matrix}$

[0158] The coefficients A₂, A₁, A₀ from (5l) can be written down in morecompact form by utilizing the following relations, which are valid forany complex numbers:

A=A ₁ +jA ₂ , B=B ₁ +jB ₂:

{A ₁}² +{A ₂}² =|A| ²  (5m)

{B ₁}² +{B ₂}² =|B| ²  (5n)

real{ AB *}=real{(A ₁ +jA ₂)(B ₁ −jB ₂)}=real{(A ₁ B ₁ +A ₂ B ₂)+j(A ₂ B₁ −A ₁ B ₂)}=A ₁ B ₁ +A ₂ B ₂  (5o)

[0159]X* denotes conjugate of X

[0160] |X| denotes absolute value of X.

[0161] Finally, taking into account the above relations for complexnumbers (5m)-(5o), the quadratic algebraic equation for the faultdistance (d)-formula (5l) transforms to the formula (6).

[0162] APPENDIX 2—Derivation of the Formula Determining the Value Of theSynchronization Angle (δ_(m))-equation (A1_(—)9)

[0163] The value of the synchronization angle (δ_(m)) is calculated fromthe following condition for the circuit of FIG. 5:

I _(Ax) =−I _(Bx)  (A1_(—)1)

[0164] where:

I _(Ax) =I _(A) _(—) _(pre) _(—) ₁ e ^(jδ) ^(_(m)) −j0.5ω₁ C ₁ V _(A)_(—) _(pre) _(—) ₁ e ^(jδ) ^(_(m))

I _(Bx) =I _(B) _(—) _(pre) _(—) ₁ −j0.5ω₁ C ₁ V _(B) _(—) _(pre) _(—) ₁

[0165] From (A1_(—)1) one obtains: $\begin{matrix}{^{j\quad \delta_{m}} = {\frac{{- {\underset{\_}{I}}_{{B\_ pre}\_ 1}} + {j\quad 0.5\quad \omega_{1}C_{1}{\underset{\_}{V}}_{{B\_ pre}\_ 1}}}{{\underset{\_}{I}}_{{A\_ pre}\_ 1} - {j\quad 0.5\quad \omega_{1}C_{1}{\underset{\_}{V}}_{{A\_ pre}\_ 1}}} = {P + {j\quad Q}}}} & \left( {{A1\_}2} \right) \\{\delta_{m} = {{{angle}\left( {P + {j\quad Q}} \right)} = {\tan^{- 1}\left( \frac{Q}{P} \right)}}} & \left( {{A1\_}3} \right)\end{matrix}$

[0166] To calculate the coefficients (P, Q) from (A1_(—)2)-(A1_(—)3) oneuses the following substitutions:

I _(A) _(—) _(pre) _(—) ₁ =I _(A1) +jI _(A2)  (A1_(—)4)

I _(B) _(—) _(pre) _(—) ₁ =I _(B1) +jI _(B2)  (A1_(—)5)

V _(A) _(—) _(pre) _(—) ₁ =V _(A1) +jV _(A2)  (A1_(—)6)

V _(B) _(—) _(pre) _(—) ₁ =V _(B1) +jV _(B2)  (A1_(—)7)

[0167] Thus, formula (A1_(—)2) takes the form: $\begin{matrix}\begin{matrix}{^{j\quad \delta_{m}} = \frac{{- \left( {I_{B1} + {j\quad I_{B2}}} \right)} + {{j0}{.5}\omega_{1}{C_{1}\left( {V_{B1} + {j\quad V_{B2}}} \right)}}}{\left( {I_{A1} + {j\quad I_{A2}}} \right) - {{j0}{.5}\omega_{1}{C_{1}\left( {V_{A1} + {j\quad V_{A2}}} \right)}}}} \\{= \frac{{- \left( {I_{B1} - {0.5\omega_{1}C_{1}V_{B2}}} \right)} + {j\left( {{- I_{B2}} + {0.5\omega_{1}C_{1}V_{B1}}} \right)}}{\left( {I_{A1} + {0.5\omega_{1}C_{1}V_{A2}}} \right) + {j\left( {I_{A2} - {0.5\omega_{1}C_{1}V_{A1}}} \right)}}} \\{= \frac{N_{1} + {j\quad N_{2}}}{D_{1} + {j\quad D_{2}}}} \\{= \frac{\left( {N_{1} + {j\quad N_{2}}} \right)\left( {D_{1} - {j\quad D_{2}}} \right)}{\left( {D_{1} + {j\quad D_{2}}} \right)\left( {D_{1} - {j\quad D_{2}}} \right)}} \\{= \frac{\left( {{N_{1}D_{1}} + {N_{2}D_{2}}} \right) + {j\left( {{N_{2}D_{1}} - {N_{1}D_{2}}} \right)}}{D_{1}^{2} + D_{2}^{2}}} \\{= {{\frac{{N_{1}D_{1}} + {N_{2}D_{2}}}{D_{1}^{2} + D_{2}^{2}} + {j\frac{{N_{2}D_{1}} - {N_{1}D_{2}}}{D_{1}^{2} + D_{2}^{2}}}} = {P + {j\quad Q}}}}\end{matrix} & \left( {{A1\_}8} \right)\end{matrix}$

[0168] In consequence, the formula (A1_(—)3) for the synchronizationangle (δ_(m)) takes the form: $\begin{matrix}{\delta_{m} = {{{angle}\left( {P + {j\quad Q}} \right)} = {{\tan^{- 1}\left( \frac{Q}{P} \right)} = {\tan^{- 1}\left( \frac{{N_{2}D_{1}} - {N_{1}D_{2}}}{{N_{1}D_{1}} + {N_{2}D_{2}}} \right)}}}} & \left( {{A1\_}9} \right)\end{matrix}$

[0169] where:

N ₁ =−I _(B1)−0.5ω₁ C ₁ V _(B2)=−real( I _(B) _(—) _(pre) _(—) ₁)−0.5ω₁C ₁imag( V _(B) _(—) _(pre) _(—) ₁)

N ₂ =−I _(B2)+0.5ω₁ C ₁ V _(B1)=−imag( I _(B) _(—) _(pre) _(—1) ₁)+0.5ω₁C ₁real( V _(B) _(—) _(pre) _(—) ₁)

D ₁ =I _(A1)+0.5ω₁ C ₁ V _(A2)=real( I _(A) _(—) _(pre) _(—) ₁)+0.5ω₁ C₁imag( V _(A) _(—) _(pre) _(—) ₁)

D ₂ =I _(A2)−0.5ω₁ C ₁ V _(A1)=imag( I _(A) _(—) _(pre) _(—) ₁)−0.5ω₁ C₁real( V _(A) _(—) _(pre) _(—) ₁)

[0170] APPENDIX 3—Derivation of the Formula for Determining the FaultPath Impedance in Case of Three-Phase Balanced Faults [Equations:(9)-(10)]

[0171] Taking the left-hand side loop (LHS) in the circuit of FIG. 4 thefollowing voltage formula can be written down:

V _(A1) e ^(jδ) −dZ _(L1) I _(A1) e ^(jδ) −Z _(F)( I _(A1) e ^(jδ) +I_(B1))=0  (a)

[0172] Thus, the fault path impedance can be calculated as:$\begin{matrix}\begin{matrix}{{\underset{\_}{Z}}_{F} = {\frac{{{\underset{\_}{V}}_{A1}^{j\quad \delta}} - {d\quad {\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}^{j\quad \delta}}}{{{\underset{\_}{I}}_{A1}^{j\quad \delta}} + {\underset{\_}{I}}_{B1}} = {\frac{{\underset{\_}{V}}_{A1} - {d\quad {\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}}}{{\underset{\_}{I}}_{A1} + \frac{{\underset{\_}{I}}_{B1}}{^{j\delta}}} = \frac{{\underset{\_}{V}}_{A1} - {d\quad {\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}}}{{\underset{\_}{I}}_{A1} + \frac{{\underset{\_}{I}}_{B1}}{\frac{{\underset{\_}{V}}_{B1} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} + {d{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}}}{{\underset{\_}{V}}_{A1} - {d\quad {\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}}}}}}}} \\{= {\frac{{\underset{\_}{V}}_{A1} - {d\quad {\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}}}{{\underset{\_}{I}}_{A1} + \frac{{\underset{\_}{I}}_{B1}\left( {{\underset{\_}{V}}_{A1} - {d\quad {\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}}} \right)}{{\underset{\_}{V}}_{B1} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} + {d{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}}}} = \frac{{\underset{\_}{V}}_{A1} - {d\quad {\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}}}{\frac{{{\underset{\_}{I}}_{A1}\left( {{\underset{\_}{V}}_{B1} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} + {d{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}}} \right)} + {{\underset{\_}{I}}_{B1}\left( {{\underset{\_}{V}}_{A1} - {d{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}}} \right)}}{{\underset{\_}{V}}_{B1} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} + {d{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}}}}}} \\{= \frac{\left( {{\underset{\_}{V}}_{A1} - {d\quad {\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}}} \right)\left( {{\underset{\_}{V}}_{B1} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} + {d{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}}} \right)}{{{\underset{\_}{I}}_{A1}\left( {{\underset{\_}{V}}_{B1} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} + {d{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}}} \right)} + {{\underset{\_}{I}}_{B1}\left( {{\underset{\_}{V}}_{A1} - {d{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}}} \right)}}} \\{= \frac{{{- {\underset{\_}{Z}}_{L1}^{2}}{\underset{\_}{I}}_{A1}{\underset{\_}{I}}_{B1}d^{2}} + {\left( {{{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}{\underset{\_}{V}}_{A1}} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}{\underset{\_}{V}}_{B1}} + {{\underset{\_}{Z}}_{L1}^{2}{\underset{\_}{I}}_{A1}{\underset{\_}{I}}_{B1}}} \right)d} + \left( {{{\underset{\_}{V}}_{A1}{\underset{\_}{V}}_{B1}} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}{\underset{\_}{V}}_{A1}}} \right)}{{{\underset{\_}{I}}_{A1}{\underset{\_}{V}}_{B1}} + {{\underset{\_}{I}}_{B1}{\underset{\_}{V}}_{A1}} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}{\underset{\_}{I}}_{B1}}}} \\{= {{\frac{{- {\underset{\_}{Z}}_{L1}^{2}}{\underset{\_}{I}}_{A1}{\underset{\_}{I}}_{B1}}{{{\underset{\_}{I}}_{A1}{\underset{\_}{V}}_{B1}} + {{\underset{\_}{I}}_{B1}{\underset{\_}{V}}_{A1}} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}{\underset{\_}{I}}_{B1}}}d^{2}} + {\frac{{{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}{\underset{\_}{V}}_{A1}} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}{\underset{\_}{V}}_{B1}} + {{\underset{\_}{Z}}_{L1}^{2}{\underset{\_}{I}}_{A1}{\underset{\_}{I}}_{B1}}}{{{\underset{\_}{I}}_{A1}{\underset{\_}{V}}_{B1}} + {{\underset{\_}{I}}_{B1}{\underset{\_}{V}}_{A1}} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}{\underset{\_}{I}}_{B1}}}d} +}} \\{\frac{{{\underset{\_}{V}}_{A1}{\underset{\_}{V}}_{B1}} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}{\underset{\_}{V}}_{A1}}}{{{\underset{\_}{I}}_{A1}{\underset{\_}{V}}_{B1}} + {{\underset{\_}{I}}_{B1}{\underset{\_}{V}}_{A1}} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}{\underset{\_}{I}}_{B1}}}} \\{= {{P_{2}d^{2}} + {P_{1}d} + P_{0}}}\end{matrix} & \left( b \right.\end{matrix}$

[0173] where:${\underset{\_}{P}}_{2} = \frac{{- {\underset{\_}{Z}}_{L1}^{2}}{\underset{\_}{I}}_{A1}{\underset{\_}{I}}_{B1}}{{{\underset{\_}{I}}_{A1}{\underset{\_}{V}}_{B1}} + {{\underset{\_}{I}}_{B1}{\underset{\_}{V}}_{A1}} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}{\underset{\_}{I}}_{B1}}}$${\underset{\_}{P}}_{1} = \frac{{{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}{\underset{\_}{V}}_{A1}} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}{\underset{\_}{V}}_{B1}} + {{\underset{\_}{Z}}_{L1}^{2}{\underset{\_}{I}}_{A1}{\underset{\_}{I}}_{B1}}}{{{\underset{\_}{I}}_{A1}{\underset{\_}{V}}_{B1}} + {{\underset{\_}{I}}_{B1}{\underset{\_}{V}}_{A1}} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}{\underset{\_}{I}}_{B1}}}$${\underset{\_}{P}}_{0} = \frac{{{\underset{\_}{V}}_{A1}{\underset{\_}{V}}_{B1}} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}{\underset{\_}{V}}_{A1}}}{{{\underset{\_}{I}}_{A1}{\underset{\_}{V}}_{B1}} + {{\underset{\_}{I}}_{B1}{\underset{\_}{V}}_{A1}} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}{\underset{\_}{I}}_{B1}}}$

[0174] Now, it will be shown that identical formula for the fault pathimpedance (as in (b)) is obtained when taking the right-hand side loop(RHS) in the circuit of FIG. 4. For this mesh the following voltageformula can be written down:

V _(B1)−(1−d) Z _(L1) I _(B1) −Z _(F)( I _(A1) e ^(jδ) +I _(B1))=0  (c)

[0175] The fault path impedance can be determined from (c) as:$\begin{matrix}{Z_{F} = {\frac{{\underset{\_}{V}}_{B1} - {\left( {1 - d} \right){\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}}}{{{\underset{\_}{I}}_{A1}^{j\delta}} + {\underset{\_}{I}}_{B1}} = {\frac{{\underset{\_}{V}}_{B1} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} + {d{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}}}{{{\underset{\_}{I}}_{A1}^{j\delta}} + {\underset{\_}{I}}_{B1}}\quad = {\frac{{\underset{\_}{V}}_{B1} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} + {d{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}}}{{{\underset{\_}{I}}_{A1}\frac{{\underset{\_}{V}}_{B1} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} + {d{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}}}{{\underset{\_}{V}}_{A1} - {d{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}}}} + {\underset{\_}{I}}_{B1}}\quad = {\frac{{\underset{\_}{V}}_{B1} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} + {d{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}}}{\frac{{\underset{\_}{I}}_{A1}\left( {{\underset{\_}{V}}_{B1} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} + {d{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}}} \right)}{{\underset{\_}{V}}_{A1} - {d{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}}} + \frac{{\underset{\_}{I}}_{B1}\left( {{\underset{\_}{V}}_{A1} - {d{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}}} \right)}{{\underset{\_}{V}}_{A1} - {d{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}}}}\quad = {\frac{\left( {{\underset{\_}{V}}_{B1} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} + {d{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}}} \right)\left( {{\underset{\_}{V}}_{A1} - {d{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}}} \right)}{{{\underset{\_}{I}}_{A1}\left( {{\underset{\_}{V}}_{B1} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} + {d{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}}} \right)} + {{\underset{\_}{I}}_{B1}\left( {{\underset{\_}{V}}_{A1} - {d{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}}} \right)}}\quad = {\frac{\begin{matrix}{{{- {\underset{\_}{Z}}_{L1}^{2}}{\underset{\_}{I}}_{A1}{\underset{\_}{I}}_{B1}d^{2}} +} \\{{\left( {{{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}{\underset{\_}{V}}_{A1}} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}{\underset{\_}{V}}_{B1}} + {{\underset{\_}{Z}}_{L1}^{2}{\underset{\_}{I}}_{A1}{\underset{\_}{I}}_{B1}}} \right)d} +} \\\left( {{{\underset{\_}{V}}_{A1}{\underset{\_}{V}}_{B1}} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}{\underset{\_}{V}}_{A1}}} \right)\end{matrix}}{{{\underset{\_}{I}}_{A1}{\underset{\_}{V}}_{B1}} + {{\underset{\_}{I}}_{B1}{\underset{\_}{V}}_{A1}} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}{\underset{\_}{I}}_{B1}}}\quad = {{{\frac{{\underset{\_}{Z}}_{L1}^{2}{\underset{\_}{I}}_{A1}{\underset{\_}{I}}_{B1}}{{{\underset{\_}{I}}_{A1}{\underset{\_}{V}}_{B1}} + {{\underset{\_}{I}}_{B1}{\underset{\_}{V}}_{A1}} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}{\underset{\_}{I}}_{B1}}}d^{2}} + {\frac{{{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}{\underset{\_}{V}}_{A1}} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}{\underset{\_}{V}}_{B1}} + {{\underset{\_}{Z}}_{L1}^{2}{\underset{\_}{I}}_{A1}{\underset{\_}{I}}_{B1}}}{{{\underset{\_}{I}}_{A1}{\underset{\_}{V}}_{B1}} + {{\underset{\_}{I}}_{B1}{\underset{\_}{V}}_{A1}} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}{\underset{\_}{I}}_{B1}}}d} + \frac{{{\underset{\_}{V}}_{A1}{\underset{\_}{V}}_{B1}} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}{\underset{\_}{V}}_{A1}}}{{{\underset{\_}{I}}_{A1}{\underset{\_}{V}}_{B1}} + {{\underset{\_}{I}}_{B1}{\underset{\_}{V}}_{A1}} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}{\underset{\_}{I}}_{B1}}}}\quad = {{P_{2}d^{2}} + {P_{1}d} + P_{0}}}}}}}}}} & (d)\end{matrix}$

[0176] where:${\underset{\_}{P}}_{2} = \frac{{- {\underset{\_}{Z}}_{L1}^{2}}{\underset{\_}{I}}_{A1}{\underset{\_}{I}}_{B1}}{{{\underset{\_}{I}}_{A1}{\underset{\_}{V}}_{B1}} + {{\underset{\_}{I}}_{B1}{\underset{\_}{V}}_{A1}} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}{\underset{\_}{I}}_{B1}}}$${\underset{\_}{P}}_{1} = \frac{{{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}{\underset{\_}{V}}_{A1}} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}{\underset{\_}{V}}_{B1}} + {{\underset{\_}{Z}}_{L1}^{2}{\underset{\_}{I}}_{A1}{\underset{\_}{I}}_{B1}}}{{{\underset{\_}{I}}_{A1}{\underset{\_}{V}}_{B1}} + {{\underset{\_}{I}}_{B1}{\underset{\_}{V}}_{A1}} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}{\underset{\_}{I}}_{B1}}}$${\underset{\_}{P}}_{0} = \frac{{{\underset{\_}{V}}_{A1}{\underset{\_}{V}}_{B1}} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}{\underset{\_}{V}}_{A1}}}{{{\underset{\_}{I}}_{A1}{\underset{\_}{V}}_{B1}} + {{\underset{\_}{I}}_{B1}{\underset{\_}{V}}_{A1}} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}{\underset{\_}{I}}_{B1}}}$

[0177] Thus, impedance of a fault path in case of a three-phase balancedfault is calculated in the same way, regardless from which thesubstation it is seen.

1. Method for fault location in a section of at least one transmissionline comprising: measuring the voltages and currents at both ends, A andB, of the section, obtaining the phasors of the positive sequencevoltages V _(A1), V _(B1) measured at the ends A and B, respectively,obtaining the phasors of the positive sequence currents I _(A1), I _(B1)measured at the ends A and B, respectively, using an equivalent circuitdiagram for the positive sequence quantities, thereby obtaining V _(A1)e ^(jδ) −V _(B1) +Z _(L1) I _(B1) −dZ _(L1)( I _(A1) e ^(jδ) +I_(B1))=0, where Z _(L1)—impedance of the whole line for the positivesequence, d—unknown distance to fault [pu], counted from the substationA, δ—unknown synchronization angle, characterised in determining theterm exp(jδ) expressed by the formula:$^{j\delta} = \frac{{\underset{\_}{V}}_{B1} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} + {d{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}}}{{\underset{\_}{V}}_{A1} - {d{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}}}$

calculating absolute values for both sides: A ₂ d ² +A ₁ d+A ₀=0 where:d—unknown fault distance, A ₂ =|Z _(L1) I _(A1)|² −|Z _(L1) I _(B1)|² A₁=−2real{ V _(A1)( Z _(L1) I _(A1))*+( V _(B1) −Z _(L1) I _(B1))( Z_(L1) I _(B1))*}A ₀ =|V _(A1)|² −|V _(B1) −Z _(L1) I _(B1)|² X* denotesconjugate of X, |X| denotes absolute value of X solving the equation,thereby obtaining two solutions for a distance to fault (d₁, d₂), andcomparing d₁, d₂ according to 0<(d ₁ or d ₂)<1 pu.
 2. Method accordingto claim 1, wherein, if one of d₁ or d₂ is satisfied, that value istaken as the valid result.
 3. Method according to claim 1, wherein, ifboth d₁ and d₂ are satisfied, it comprises the further steps of:determining the values of the synchronization angle corresponding toboth the solutions are determined according to:$^{{j\delta}_{1}} = \frac{{\underset{\_}{V}}_{B1} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} + {d_{1}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}}}{{\underset{\_}{V}}_{A1} - {d_{1}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}}}$$^{{j\delta}_{2}} = \frac{{\underset{\_}{V}}_{B1} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} + {d_{2}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}}}{{\underset{\_}{V}}_{A1} - {d_{2}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}}}$

assuming a maximum range for the synchronization angle as(−δ_(short)÷δ_(short))=(−π/2÷π/2) and comparing d₁, d₂ according to−δ_(short)<(δ₁ or δ₂)<δ_(short).
 4. Method according to claim 3,wherein, if one of δ₁, δ₂ is satisfied, that value of a distance tofault, d₁ or d₂, is taken as the valid solution.
 5. Method according toclaim 1, wherein, if both δ₁ and δ₂ are satisfied, it comprises thefurther steps of: calculating the value of the synchronization angleδ_(m) from the positive sequence phasors of the pre-fault voltages andcurrents measured at both ends and by using a circuit admitting shuntbranches, according to I _(Ax) =−I _(Bx) where: I _(Ax) =I _(A) _(—)_(pre) _(—) ₁ e ^(jδ) ^(_(m)) −j0.5ω₁ C ₁ V _(A) _(—) _(pre) _(—) ₁ e^(jδ) ^(_(m)) I _(Bx) =I _(B) _(—) _(pre) _(—) ₁ −j0.5ω₁ C ₁ V _(B) _(—)_(pre) _(—) ₁ leading to$\delta_{m} = {{{angle}\left( {P + {jQ}} \right)} = {\tan^{- 1}\left( \frac{{N_{2}D_{1}} - {N_{1}D_{2}}}{{N_{1}D_{1}} + {N_{2}D_{2}}} \right)}}$

where: N ₁=−real( I _(B) _(—) _(pre) _(—) ₁)−0.5ω₁ C ₁imag( V _(B) _(—)_(pre) _(—) ₁)N ₂=−imag( I _(B) _(—) _(pre) _(—) ₁)+0.5ω₁ C ₁real( V_(B) _(—) _(pre) _(—) ₁)D ₁=real( I _(A) _(—) _(pre) _(—) ₁)+0.5ω₁ C₁imag( V _(A) _(—) _(pre) _(—) ₁)D ₂=imag( I _(A) _(—) _(pre) _(—)₁)−0.5ω₁ C ₁real( V _(A) _(—) _(pre) _(—) ₁) comparing the value of thesynchronization angle δ_(m) with the values of the synchronization angle(δ₁, δ₂), calculated as: $\begin{matrix}{\delta_{1} = {{{angle}\left( \frac{{\underset{\_}{V}}_{B1} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} + {d_{1}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}}}{{\underset{\_}{V}}_{A1} - {d_{1}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}}} \right)} = {{{angle}\quad \left( {P_{1} + {jQ}_{1}} \right)} = {\tan^{- 1}\left( \frac{Q_{1}}{P_{1}} \right)}}}} & \left( {{A1\_}10} \right)\end{matrix}$

where:$P_{1} = {{real}\left( \frac{{\underset{\_}{V}}_{B1} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} + {d_{1}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}}}{{\underset{\_}{V}}_{A1} - {d_{1}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}}} \right)}$

$\begin{matrix}{{Q_{1} = {{imag}\left( \frac{{\underset{\_}{V}}_{B1} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} + {d_{1}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}}}{{\underset{\_}{V}}_{A1} - {d_{1}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}}} \right)}}\text{}{and}{\delta_{2} = {{{angle}\left( \frac{{\underset{\_}{V}}_{B1} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} + {d_{2}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}}}{{\underset{\_}{V}}_{A1} - {d_{2}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}}} \right)} = {{{angle}\quad \left( {P_{2} + {jQ}_{2}} \right)} = {\tan^{- 1}\left( \frac{Q_{2}}{P_{2}} \right)}}}}} & \left( {{A1\_}11} \right)\end{matrix}$

where:$P_{2} = {{real}\left( \frac{{\underset{\_}{V}}_{B1} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} + {d_{2}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}}}{{\underset{\_}{V}}_{A1} - {d_{2}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}}} \right)}$$Q_{2} = {{imag}\left( \frac{{\underset{\_}{V}}_{B1} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} + {d_{2}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}}}{{\underset{\_}{V}}_{A1} - {d_{2}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}}} \right)}$

performing the selection of the valid result (d_(ν)) from the solutionsof the quadratic equation A₂d²+A₁d+A₀=0 for the fault distance [(d₁ ord₂) as: if |δ₂−δ_(m)|>|δ₁−δ_(m)| then d _(ν) =d ₁ else d_(ν)=d₂ end 6.Method according to claim 5, wherein using the phasors of the particularsound phase voltages and currents measured at both ends and by using acircuit admitting shunt branches, and performing the selection of thevalid result (d_(ν)) from the solutions of the quadratic equation A₂d²+A₁d+A₀=0 for the fault distance [(d₁ or d₂) as: if|δ₂−δ_(sound)|>|δ₁−δ_(sound)| then d _(ν) =d ₁ else d_(ν)=d₂ end 7.Method according to claim 1, wherein, if both δ₁ and δ₂ are satisfied,it comprises the further steps of: determining that the fault is athree-phase fault, estimating the fault path impedances, determiningwhich of the fault path impedances are closest to the condition Z_(F)=R_(F), selecting the valid result (d_(ν)) from the solutions of thequadratic equation A₂d²+A₁d+A₀=0 for the fault distance (d₁ or d₂) asfollows: if |imag( Z _(F1)(d ₁))|<|imag( Z _(F2)(d ₂))| then d _(ν) =d ₁else d_(ν)=d₂ end
 8. Method according claim 1, wherein, if both δ₁ andδ₂ are satisfied, it comprises the further steps of: using an equivalentcircuit diagram for the negative sequence for obtaining a quadraticequation for the sought fault distance (d) as: B ₂ d ² +B ₁ d+B ₀=0  (a)where: B₂, B₁, B₀ are: B ₂ =|Z _(L1) I _(A2)|² −|Z _(L1) I 1 _(B2)|² B₁=−2real{ V _(A2)( Z _(L1) I _(A2))*+( V _(B2) −Z _(L1) I _(B2))( Z_(L1) I _(B2))*}  (b)B ₀ =|V _(A2)|² −|V _(B2) −Z _(L1) I _(B2)|²solving equation (a), taking into account equation (b), therebyobtaining the next two solutions for the fault distance (d₃, d₄):$d_{3} = \frac{{- B_{1}} - \sqrt{B_{1}^{2} - {4B_{2}B_{0}}}}{2B_{2}}$$d_{4} = \frac{{- B_{1}} + \sqrt{B_{1}^{2} - {4B_{2}B_{0}}}}{2B_{2}}$

where the solutions, taken out of all four solutions (d₁, d₂, d₃, d₄),which coincide (d₁=d_(j), where: i=1 or 2, j=3 or 4) indicate the validsolution for the fault distance (d_(ν)).
 9. Method for fault location ina section of at least one transmission line comprising: measuring thevoltages and currents at both ends, A and B, of the section, obtainingthe phasors of the positive sequence voltages V _(A1), V _(B1) measuredat the ends A and B, respectively, obtaining the phasors of the positivesequence currents I _(A1), I _(B1) measured at the ends A and B,respectively, and at the occurrence of a fault, determining whether itis a 3-phase balanced fault or not, and if not, using an equivalentcircuit diagram for the negative sequence for obtaining a quadraticequation for the sought fault distance (d) as: B ₂ d ² +B ₁ d+B ₀=0  (a)where: B₂, B₁, B₀ are the real number coefficients expressed as: B ₂ =|Z_(L1) I _(A2)|² −|Z _(L1) I _(B2)|² B ₁=−2real{ V _(A2)( Z _(L1) I_(A2))*+( V _(B2) −Z _(L1) I _(B2))( Z _(L1) I _(B2))*}  (b)B ₀ =|V_(A2)|² −|V _(B2) −Z _(L1) I _(B2)|² solving equation (a), talking intoaccount equation (b), thereby obtaining the next two solutions for thefault distance (d₃, d₄):$d_{3} = \frac{{- B_{1}} - \sqrt{B_{1}^{2} - {4B_{2}B_{0}}}}{2B_{2}}$${d_{4} = \frac{{- B_{1}} + \sqrt{B_{1}^{2} - {4B_{2}B_{0}}}}{2B_{2}}},{and}$

comparing d₃, d₄ according to 0<(d ₃ or d ₄)<1 pu.
 10. Method accordingto claim 9, wherein, if one of d₃ or d₄ is satisfied, that value istaken as the valid result.
 11. Method according to claim 9, wherein, ifboth d₃ and d₄ are satisfied, it comprises the further steps of:determining the values of the synchronization angle corresponding toboth the solutions are determined according to:$^{j^{\delta_{3}}} = \frac{{\underset{\_}{V}}_{B1} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} + {d_{1}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}}}{{\underset{\_}{V}}_{A1} - {d_{3}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}}}$$^{j^{\delta_{4}}} = \frac{{\underset{\_}{V}}_{B1} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} + {d_{2}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}}}{{\underset{\_}{V}}_{A1} - {d_{4}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}}}$

assuming a maximum range for the synchronization angle as(−δ_(short)÷δ_(short))=(−π/2÷π/2) and comparing d₃, d₄ according to−δ_(short)<(δ₃ or δ₄)<δ_(short).
 12. Method according to claim 11,wherein, if one of δ₃, δ₄ is satisfied, that value of a distance tofault, d₃ or d₄, is taken as the valid solution.
 13. Method according toclaim 9, wherein, if both δ₃ and δ₄ are satisfied, it comprises thefurther steps of: calculating the value of the synchronization angleδ_(m) from the positive sequence phasors of the pre-fault voltages andcurrents measured at both ends and by using a circuit admitting shuntbranches, according to I _(Ax) =−I _(Bx) where: I _(Ax) =I _(A) _(—)_(pre) _(—) ₁ e ^(jδ) ^(_(m)) −j0.5ω₁ C ₁ V _(A) _(—) _(pre) _(—) ₁ e^(jδ) ^(_(m)) I _(Bx) =I _(B) _(—) _(pre) _(—) ₁ −j0.5ω₁ C ₁ V _(B) _(—)_(pre) _(—) ₁ leading to$\delta_{m} = {{{angle}\quad \left( {P + {jQ}} \right)} = {\tan^{- 1}\left( \frac{{N_{2}D_{1}} - {N_{1}D_{2}}}{{N_{1}D_{1}} + {N_{2}D_{2}}} \right)}}$

where: N ₁=−real( I _(B) _(—) _(pre) _(—) ₁)−0.5ω₁ C ₁imag( V _(B) _(—)_(pre) _(—) ₁)N ₂=−imag( I _(B) _(—) _(pre) _(—) ₁)+0.5ω₁ C ₁real( V_(B) _(—) _(pre) _(—) ₁)D ₁=real( I _(A) _(—) _(pre) _(—) ₁)+0.5ω₁ C₁imag( V _(A) _(—) _(pre) _(—) ₁)D ₂=imag( I _(A) _(—) _(pre) _(—)₁)−0.5ω₁ C ₁real( V _(A) _(—) _(pre) _(—) ₁) comparing the value of thesynchronization angle δ_(m) with the values of the synchronization angle(δ₃, δ₄), calculated as: $\begin{matrix}{\delta_{3} = {{{angle}\left( \frac{{\underset{\_}{V}}_{B1} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} + {d_{1}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}}}{{\underset{\_}{V}}_{A1} - {d_{1}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}}} \right)} = {{{angle}\left( {P_{1} + {jQ}_{1}} \right)} = {\tan^{- 1}\left( \frac{Q_{1}}{P_{1}} \right)}}}} & \quad\end{matrix}$

where:$P_{1} = {{real}\left( \frac{{\underset{\_}{V}}_{B1} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} + {d_{1}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}}}{{\underset{\_}{V}}_{A1} - {d_{1}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}}} \right)}$$Q_{1} = {{imag}\left( \frac{{\underset{\_}{V}}_{B1} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} + {d_{1}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}}}{{\underset{\_}{V}}_{A1} - {d_{1}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}}} \right)}$and$\delta_{4} = {{{angle}\left( \frac{{\underset{\_}{V}}_{B1} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} + {d_{2}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}}}{{\underset{\_}{V}}_{A1} - {d_{2}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}}} \right)} = {{{angle}\left( {P_{2} + {jQ}_{2}} \right)} = {\tan^{- 1}\left( \frac{Q_{2}}{P_{2}} \right)}}}$

where:$P_{2} = {{real}\left( \frac{{\underset{\_}{V}}_{B1} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} + {d_{2}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}}}{{\underset{\_}{V}}_{A1} - {d_{2}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}}} \right)}$$Q_{2} = {{imag}\left( \frac{{\underset{\_}{V}}_{B1} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} + {d_{2}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}}}{{\underset{\_}{V}}_{A1} - {d_{2}{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}}} \right)}$

performing the selection of the valid result (d_(ν)) from the solutionsof the quadratic equation A₂d²+A₁d+A₀=0 for the fault distance [(d₃ ord₄) as: if |δ₄−δ_(m)|>|δ₃−δ_(m)| then d _(ν) =d ₃ else d_(ν)=d₄ end 14.Method according to claim 13, wherein using the phasors of theparticular sound phase voltages and currents measured at both ends andby using a circuit admitting shunt branches, and performing theselection of the valid result (d_(ν)) from the solutions of thequadratic equation A₂d²+A₁d+A₀=0 for the fault distance [(d₃ or d₄) as:if |δ₄−δ_(sound)|>|δ₃−δ_(sound)| then d _(ν) =d ₃ else d_(ν)=d₄ end 15.Method according to claim 9, wherein, if both δ₃ and δ₄ are satisfied,it comprises the further steps of: determining that the fault is athree-phase fault, estimating the fault path impedances, determiningwhich of the fault path impedances are closest to the condition Z_(F)=R_(F) selecting the valid result (d_(ν)) from the solutions of thequadratic equation A₂d²+A₁d+A₀=0 for the fault distance (d₃ or d₄) asfollows: if |imag( Z _(F1)(d₃))|<|imag( Z _(F2)(d ₄))| then d _(ν) =d ₃else d_(ν)=d₄ end
 16. Method according to claim 9, wherein, if both δ₁and δ₂ are satisfied, it comprises the further steps of: using anequivalent circuit diagram for the positive sequence for obtaining aquadratic equation for the sought fault distance (d) as: A ₂ d ² +A ₁d+A ₀=0  (a) where: A ₂ =|Z _(L1) I _(A1)|² −|Z _(L1) I _(B1)|² A₁=−2real{ V _(A1)( Z _(L1) I _(A1))*+( V _(B1) −Z _(L1) I _(B1))( Z_(L1) I _(B1))*}  (b)A ₀ =|V _(A1)|² −|V _(B1) −Z _(L1) I _(B1)|²solving equation (a), taking into account equation (b), therebyobtaining the next two solutions for the fault distance (d₁, d₂):$d_{1} = \frac{{- A_{1}} - \sqrt{A_{1}^{2} - {4A_{2}A_{0}}}}{2A_{2}}$$d_{2} = \frac{{- A_{1}} + \sqrt{A_{1}^{2} - {4A_{2}A_{0}}}}{2A_{2}}$

where the solutions, taken out of all four solutions (d₁, d₂, d₃, d₄),which coincide (d_(i)=d_(j), where: i=1 or 2, j=3 or 4) indicate thevalid solution for the fault distance (d_(ν)).
 17. Method according toclaim 9, characterized in, if the fault is a 3-phase balanced fault,using an equivalent circuit diagram for incremental positive sequencequantities, wherein incremental positive sequence quantities are definedas the difference between post-fault and pre-fault values, as A _(2inc)d ² +A _(1inc) d+A _(0inc)=0  (a) where: A _(2inc) =|Z _(L1inc) I_(A1inc)|² −|Z _(L1inc) I _(B1inc)|² A _(1inc)=−2real{ V _(A1inc)( Z_(L1inc) I _(A1inc))*+( V _(B1inc) −Z _(L1inc) I _(B1inc))( Z _(L1inc) I_(B1inc))*}  (b)A _(0inc) =|V _(A1inc)|² −|V _(B1inc) −Z _(L1inc) I_(B1inc)|² solving equation (a), taking into account equation (b),thereby obtaining two solutions for the fault distance (d₅, d₆) andcomparing d₅, d₆ according to 0<(d ₅ or d ₆)<1 pu.
 18. Method accordingto claim 17, wherein, if one of d₅ or d₆ is satisfied, that value istaken as the valid result.
 19. Method according to claim 17, wherein, ifboth d₅ and d₆ are satisfied, it comprises the further steps of any ofthe claims 11-16.
 20. Method according to any of the preceding claims,further comprising compensation of shunt capacitances, comprising thesteps of: performing the compensation for the positive sequence currentsaccording to: I _(A1) _(—) _(comp) _(—) ₁ =I _(A1) −j0.5ω₁ C ₁ d _(ν) V_(A1)  (14) I _(B1) _(—) _(comp) _(—) ₁ =I _(B1) −j0.5ω₁ C ₁(1−d _(ν)) V_(B1)  (15) where: d_(ν)—fault distance (the valid result) obtainedwithout taking into account the shunt capacitances of a line, C₁—shuntcapacitance of a whole line, taking the phasors compensated for shuntcapacitances and transforming the quadratic algebraic equation for afault distance before the compensation to the following quadraticalgebraic equation for the improved fault distance: A ₂ _(comp) _(—) ₁ d² _(comp) _(—) ₁ +A ₁ _(—) _(comp) _(—) ₁ d _(comp) _(—) ₁ +A ₀ _(—)_(comp) _(—) ₁=0  (a) where: d_(comp) _(—) ₁—the improved fault distanceresult (result of the first iteration of the compensation), A₂ _(—)_(comp) _(—) ₁, A₁ _(—) _(comp) _(—) ₁, A₀ _(—) _(comp) _(—)₁—coefficients (real numbers) expressed duly with the unsynchronizedmeasurements at the line terminals and with the positive sequenceimpedance of a line (Z _(L1)); using the measurements the originalpositive sequence phasors of voltages (V _(A1), V _(B1)) and the phasorsof currents compensated for a line shunt capacitances (I _(A1) _(—)_(comp) _(—) ₁, I _(B1) _(—) _(comp) _(—) ₁): A ₂ _(—) _(comp) _(—) ₁=|Z _(L1) I _(A1) _(—) _(comp) _(—) ₁|² −|Z _(L1) I _(B1) _(—) _(comp)_(—) ₁|² A ₁ _(—) _(comp) _(—) ₁=−2real{ V _(A1)( Z _(L1) I _(A1) _(—)_(comp) _(—) ₁)*+( V _(B1) −Z _(L1) I _(B1) _(—) _(comp) _(—) ₁)( Z_(L1) I _(B1) _(—) _(comp) _(—) ₁)*}A ₀ _(—) _(comp) _(—) ₁ =|V _(A1)|²−|V _(B1) −Z _(L1) I _(B1) _(—) _(comp) _(—) ₁|²  (b) solving equation(a), and taking (b), thereby obtaining the two solutions for the faultdistance (d_(comp) _(—) _(1A), d_(comp) _(—) _(1B)):$d_{{comp\_}1A} = \frac{{- A_{1{\_ comp}\_ 1}} - \sqrt{A_{1{\_ comp}\_ 1}^{2} - {4A_{2{\_ comp}\_ 1}A_{0{\_ comp}\_ 1}}}}{2A_{2{\_ comp}\_ 1}}$$d_{{comp\_}1B} = \frac{{- A_{1{\_ comp}\_ 1}} + \sqrt{A_{1{\_ comp}\_ 1}^{2} - {4A_{2{\_ comp}\_ 1}A_{0{\_ comp}\_ 1}}}}{2A_{2{\_ comp}\_ 1}}$

obtaining the improved value for the fault distance (d_(ν) _(—) _(comp)_(—) ₁) as a result of the first iteration of the compensation,selection of a particular solution according to: if before thecompensation the solution d₁ was selected as the valid result(d_(ν)=d₁), then d_(comp) _(—) _(1A) is taken as the valid result afterthe first iteration of the compensation (d_(ν) _(—) _(comp) _(—)₁=d_(comp) _(—) _(1A)), if, before the compensation the solution d₂ wasselected as the valid result (d_(ν)=d₂) then d_(comp) _(—) _(1B) istaken as the valid result after the first iteration of the compensation(d_(ν) _(—) _(comp) _(—) ₁=d_(comp) _(—) _(1B)), continue the iterationsuntil the convergence with the pre-defined margin (d_(margin)) isachieved: |d _(ν) _(—) _(comp) _(—) ₁ −d _(ν) _(—) _(comp) _(—) _(i−1)|<d _(margin) where: the index (i) is the present iteration while theindex (i−1) is the preceding iteration.
 21. Device for fault location ina section of at least one transmission line comprising: means formeasuring the voltages and currents at both ends, A and B, of thesection, means for obtaining the phasors of the positive sequencevoltages V _(A1), V _(B1) measured at the ends A and B, respectively,means for obtaining the phasors of the positive sequence currents I_(A1), I _(B1) measured at the ends A and B, respectively, using anequvalent circuit diagram for the positive sequence quantities, therebyobtaining V _(A1) e ^(jδ) −V _(B1) +Z _(L1) I _(B1) −dZ _(L1)( I _(A1) e^(jδ) +I _(B1))=0, where Z _(L1)—impedance of the whole line for thepositive sequence, d—unknown distance to fault [pu], counted from thesubstation A, δ—unknown synchronization angle, characterised in meansfor determining the term exp(jδ) expressed by the formula:$^{j^{\delta}} = \frac{{\underset{\_}{V}}_{B1} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} + {d{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}}}{{\underset{\_}{V}}_{A1} - {d{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}}}$

means for calculating absolute values for both sides: A ₂ d ² +A ₁ d+A₀=0 where: d—unknown fault distance,$A_{2} = {{{{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}}}^{2} - {{{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}}}^{2}}$$A_{1} = {{- 2}{real}\left\{ {{{\underset{\_}{V}}_{A1}\left( {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{A1}} \right)}^{*} + {\left( {{\underset{\_}{V}}_{B1} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}}} \right)\left( {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}} \right)^{*}}} \right\}}$$A_{0} = {{{\underset{\_}{V}}_{A1}}^{2} - {{{\underset{\_}{V}}_{B1} - {{\underset{\_}{Z}}_{L1}{\underset{\_}{I}}_{B1}}}}^{2}}$

X* denotes conjugate of X, |X| denotes absolute value of X means forsolving the equation, thereby obtaining two solutions for a distance tofault (d₁, d₂), and comparing d₁, d₂ according to 0<(d ₁ or d ₂)<1 pu.22. Device for fault location in a section of at least one transmissionline comprising: means for measuring the voltages and currents at bothends, A and B, of the section, means for obtaining the phasors of thepositive sequence voltages V _(A1), V _(B1) measured at the ends A andB, respectively, means for obtaining the phasors of the positivesequence currents I _(A1), I _(B1) measured at the ends A and B,respectively, and at the occurrence of a fault, means for determiningwhether it is a 3-phase balanced fault or not, and if not, means forusing an equivalent circuit diagram for the negative sequence forobtaining a quadratic equation for the sought fault distance (d) as: B ₂d ² +B ₁ d+B ₀=0  (a) where: B₂, B₁, B₀ are the real number coefficientsexpressed as: B ₂ =|Z _(L1) I _(A2)|² −|Z _(L1) I _(B2)|² B ₁=−2real{ V_(A2)( Z _(L1) I _(A2))*+( V _(B2) −Z _(L1) I _(B2))( Z _(L1) I_(B2))*}  (b)B ₀ =|V _(A2)|² −|V _(B2) −Z _(L1) I _(B2)|² means forsolving equation (a), taking into account equation (b), therebyobtaining the next two solutions for the fault distance (d₃, d₄):$d_{3} = \frac{{- B_{1}} - \sqrt{B_{1}^{2} - {4B_{2}B_{0}}}}{2B_{2}}$${d_{4} = \frac{{- B_{1}} + \sqrt{B_{1}^{2} - {4B_{2}B_{0}}}}{2B_{2}}},{and}$

means for comparing d₃, d₄ according to 0<(d ₃ or d ₄)<1 pu.
 23. Deviceaccording to claim 22, characterized in, if the fault is a 3-phasebalanced fault, means for using an equivalent circuit diagram forincremental positive sequence quantities, wherein incremental positivesequence quantities are defined as the difference between post-fault andpre-fault values, as A _(2inc) d ² +A _(1inc) d+A _(0inc)=0  (a) where:A _(2inc) =|Z _(L1inc) I _(A1inc)|² −|Z _(L1inc) I _(B1inc)|² A_(1inc)=−2real{ V _(A1inc)( Z _(L1inc) I _(A1inc))*+( V _(B1inc) −Z_(L1inc) I _(B1inc))( Z _(L1inc) I _(B1inc))*}  (b)V _(0inc) =|V_(A1inc)|² −|V _(B1inc) −Z _(L1inc) I _(B1inc)|² means for solvingequation (a), taking into account equation (b), thereby obtaining twosolutions for the fault distance (d₅, d₆), and comparing d₅, d₆according to 0<(d ₅ or d₆)<1 pu.
 24. Use of a device according to claim21 or 22 to determine the distance to fault in a single transmissionline.
 25. Use of a device according to claim 21 or 22 to determine thedistance to fault in parallel mutually coupled transmission line. 26.Use of a device according to claim 21 or 22 to determine the distance tofault in a distribution line.
 27. A computer data signal embodied in adata communication comprising information about a characteristic of atransmission line or distribution line, characterised in that saidsignal is sent by a system for fault location over a communicationsnetwork and includes information about a distance (d) to a faultcalculated according to any of the claims 1-20 such that upon receipt ofsaid signal a control action may be enabled in respect of said fault.28. Computer program product comprising computer code means and/orsoftware code portions for making a computer or processor perform thesteps of any of the claims 1-20.
 29. Computer program product accordingto claim 28, stored on a computer-readable medium.